Stable central limit theorems for discrete-time lag martingale difference arrays

2025/10/07 by Walter Dempsey, Dempsey, Walter, Easton Huch +1
Decision Sciences · Mathematics · #60B12 (Secondary) #60F05 (Primary) #60G42 #60G48 #FOS: Mathematics #Nonlinear Differential Equations Analysis #Probability (math.PR) #Probability and Risk Models #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2510.06524

openalex publication_date 2025/10/07 · openalex created_date 2025/10/13 · openalex updated_date 2026/07/28

Abstract

Recent work in dynamic causal inference introduced a class of discrete-time stochastic processes that generalize martingale difference sequences and arrays as follows: the random variates in each sequence have expectation zero given certain lagged filtrations but not given the natural filtration. We formalize this class of stochastic processes and prove a stable central limit theorem (CLT) via a Bernstein blocking scheme and an application of the classical martingale CLT. We generalize our limit theorem to vector-valued processes via the Cramér-Wold device and develop a simple form for the limiting variance. We demonstrate the application of these results to a problem in dynamic causal inference and present a simulation study supporting their validity.

Citations

• On stable central limit theorems for multivariate discrete-time martingales
• Panel Experiments and Dynamic Causal Effects: A Finite Population\n Perspective
• Doubly Robust Identification for Causal Panel Data Models
• Design-based Analysis in Difference-In-Differences Settings with Staggered Adoption
• Two-Way Fixed Effects Estimators with Heterogeneous Treatment Effects
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